To obtain our first and simplest latent variable model, we specify a distribution to the latent vector since the lower dimensional encoded message is unobservable. Assigning a standard Gaussian to the latent variables and restricting the covariance of the errors to be isotropic gives us probabilistic PCA [1]}. The model can be expressed as
\(\prod _{n=1}^N \mathcal {N}_p( \mathbf {y}_n \mid \mathbf {W}\mathbf {z}_n, \ \sigma ^2\mathbf {I}) \mathcal {N}_d(\mathbf {z}_n \mid 0,\,\mathbf {I})\) 
